\section{Introduction}

\subsection{What is Gillies' Conditional?}
The ``Gillies Conditional'' originates with \cite{gillies_epistemic_2004},
``Epistemic conditionals and conditional Epistemics''

Unlike \emph{material implication} or the \emph{counterfactual conditional},
Gillies' conditional more directly engages the Ramsey test and epistemology,
trying to model the internal process one employs when deliberating  on
questions like ``If I knew that $X$, would I know that $Y$''?

Forms of this conditional appear informally in \cite{stalnaker_theory_1968}
and \cite{karttunen_syntax_1977}.  The first formal appearance we are aware of
is in \cite{heim_semantics_1982}.\footnote{We'd like to thank Dr. Katrin Schulz
for bringing this earlier work on Gillies' epistemic indicative to our
attention.}

However, we shall adhere to Gillies' presentation.  Gillies motivates thinking
with a thought experiment\ldots

\subsection{Motivation}

In \cite{gillies_epistemic_2004} we hear the following story:
\begin{quote}
  [Suppose] that there has been a murder in the mansion. The mansion staff can
  be partitioned into the grounds staff (people who work outside the mansion)
  and the house staff (people who work inside the mansion proper). Again, the
  three suspects are the driver (grounds staff), the butler (house staff), and
  the gardener (grounds staff). You are the lead investigator, and your young
  assistant (who, in fact, is but an apprentice) has been collecting clues and
  reports back to you. She has collected this one clue: the butler has an
  airtight alibi. Your assistant thus decrees:
\end{quote}

\begin{quote}
  \newcounter{gilliesex} % sexual...
  \begin{list}{\textup{(\arabic{gilliesex})}}{\usecounter{gilliesex}} 
    \item \emph{Therefore: if a member of the grounds staff did it, then it was
      the driver.}
  \end{list}

  Being a seasoned inspector, you disagree:

  \begin{list}{\textup{(\arabic{gilliesex})}}{\usecounter{gilliesex}}
    \setcounter{gilliesex}{1}
    \item\label{gsex2} \emph{It's not so that if a member of the grounds staff
      did it, then it was the driver. After all, it might still be the gardener
      who did it.}
  \end{list}

  The truth-functional analysis of conditional statements (i.e., the material
  conditional) treats an indicative \emph{If $p$, then $q$} as equivalent to the
  disjunction \emph{not-$p$ or $q$}. This cannot be right. If the
  truth-functional analysis were right, then your denial in \eqref{gsex2} would
  commit you to accepting that a member of the grounds staff did it and it was
  definitely not the driver.
\end{quote}

\subsubsection{Veltman's Mantra}
One way to avoid the above problem is not to commit to a truth functional
definition of validity when thinking about epistemic indicatives.
\begin{quote}
  Mantra: You know the meaning of a sentence if you know the change it brings
  about in the information state of anyone who accepts the news conveyed by it.
  \cite{veltman_defaults_1996}
\end{quote}

\ldots the conditional that Gillies proposes, based on his thought experiment
uses Veltman style epistemic update semantics.

\subsection{Syntax \& Semantics}\label{syntaxandsemantics}
\subsubsection{Review of Update Semantics}
To give a formal account of Gillies' conditional, we review (and slightly adapt)
the presentation in \cite{van_der_does_updatemight_1997}.

We are going to be concerned with the following grammars, given in
\emph{Backus-Naur form}, where $\Phi$ is a set of proposition letters:
\begin{itemize}
  \item $\mathcal{L}_0$, the language of \emph{propositional logic}:
    \[ \phi_0\ {::=}\ p \in \Phi \ |\ \bot \ |\ \neg \phi_0 \ |\ 
       \phi_0 \wedge \psi_0 \]
  \item $\mathcal{L}_\Box$, the language of \emph{$\Box$}:
    \[ \phi_\Box\ {::=}\ \phi_0 \ | \ \Box \phi_0  \]
  \item $\mathcal{L}_\gillies$, the language of \emph{Gillies' conditional}:
    \[ \phi_\gillies\ {::=}\ \phi_0 \ | \ \phi_0 \gillies \phi_0 \]
\end{itemize}
 
Abstract Semantics: Let $\mathbb{A}\ {:=}\  \langle A, \mathbf{1}, \mathbf{0},
\sleq , \sqcap , \sim, \ll\cdot\rr \rangle$ be a \emph{Boolean lattice} with
operators $\ll\cdot\rr : A \to \Phi \to A$, which employ \emph{reverse Polish
notation} and conform to the following axioms:
\begin{itemize}
  \item \emph{Reflection}: $\alpha \ll p \rr \sleq \alpha$ for all $\alpha \in A$
    and $p \in \Phi$
  \item \emph{Monotony}: If $\alpha \sleq \beta$ then $\alpha \ll p \rr \sleq
    \beta \ll p \rr$ for all $\{ \alpha, \beta\} \subseteq A$ and $p \in \Phi$
  \item \emph{Introspection}: If $\alpha \sleq \beta\ll p \rr $ then $\alpha  =
    \alpha \ll p \rr$ for all $\{ \alpha, \beta\} \subseteq A$ and $p \in \Phi$
\end{itemize} 

Using the primitive operator $\ll \cdot \rr$, define another reverse Polish
operator $[\cdot] : A \to \mathcal{L}_0 \cup \mathcal{L}_\gillies \cup
\mathcal{L}_\Box \to A$ recursively as follows, which is referred to as
\emph{epistemic update}:
\begin{itemize}
  \item $\alpha[p]\ {:=}\ \alpha\ll p\rr $
  \item $\alpha[\bot]\ {:=}\ \mathbf{0} $
  \item $\alpha[\neg \phi]\ {:=}\ \alpha \sqcap \sim \alpha[\phi] $
  \item $\alpha[\phi \wedge \psi]\ {:=}\ \alpha[\phi] \sqcap
    \alpha[\psi] $
  \item $\alpha[\Box \psi]\ {:=}\ \begin{cases} \alpha &
    \textup{if } \alpha[\phi]=\alpha\\
    \textbf{0} & \textup{otherwise}\end{cases} $
  \item $\alpha[\phi \gillies \psi]\ {:=}\ \begin{cases} \alpha &
    \textup{if } \alpha[\phi][\psi]=\alpha[\phi]\\
    \textbf{0} & \textup{otherwise}\end{cases} $
\end{itemize}

\subsubsection{Intuition}
Why is Gillies' epistemic indicative conditional defined like this? Gillies
offers some intuition in \cite{gillies_epistemic_2004}:

\begin{quote}
  I should believe an indicative ``If $p$ then $q$'' just in case learning $p$
  given my present information would be enough to commit me to $q$. Epistemic
  conditionals seem to tell us more about the structure of our information about
  the world than they do (directly, anyway) about the world.
\end{quote}

\subsubsection{Why Must?}
$\Box$ and Gillies' conditional are intimately linked.

We found that in doing formal work with the Gillies' conditional, it was
convenient to elucidate and capitalize on this relationship.

In the subsequent section, we shall investigate certain key lemmas linking $\Box$
and $\gillies$.

